We show that for nonnegative plurisubharmonic functions on the unit ball of $C^{n}$ the admissible maximal functions are dominated by the radial maximal functions in $L^{p}$-mean. This gives another characterization of the class $M^{p}$ of holomorphic functions and its invariance under the compositions by automorphisms of the unit ball. As a consequence of the invariance all onto endomorphisms of $M^{1}$ $(n=1)$ are characterized.

We show that the category $\mathbf{Conv_{B}}$ of convergence spaces over $B$ is a convenient category for any $B \in \mathbf{Conv}$ . It is shown that without any condition on spaces the category $\mathbf{Conv_{B}}$ and the category $\mathbf{Conv_{B}^{B}}$ of sectioned convergence spaces over $B$ hold various exponential laws in a natural way. In $\mathbf{Conv_{B}}$, we can construct exponential object in terms of function spaces. Our fibrewise mapping space structure generalizes the fibrewise compact-open topology in some case.

For any given approximate resolution $p=\{p_{a}|a\in A\}:X \rightarrow \mathscr{X}=(X_{a}, \mathscr{U}_{a}, p_{aa^{\prime}}, A)$ of a topological space $X$, where $\mathscr{X}$ is uniform, all $X_{a}$ are paracompact, all $\mathscr{U}_{a}$ are locally finite and $A$ is cofinite, and any given compact Hausdorff space $Y$, the approximate resolution $r=p\times 1=\{r_{b}=p_{a}\times 1|b=(a, \varphi)\in B\}$: $X \times Y \rightarrow \mathscr{X} \times Y =(X_{a}\times Y, \mathscr{U}_{a}\times\varphi[\mathscr{U}_{a}], p_{aa^{\prime}}\times 1, B)$ of the product space $X\times Y$ is constructed. Here, the indexing set $B$ is obtained by means of the set $A$ and certain subfamilies of $\Phi(a)=\{\varphi|\varphi:\mathscr{U}_{a}\rightarrow Cov(Y)\}, a\in A$, while the mesh $\mathscr{U}_{a} \times \varphi[\mathscr{U}_{a}]$ is a stacked covering of $X_{a}\times Y$ over $\mathscr{U}_{a}$.